Sinc function

In mathematics, physics, and engineering, the sinc function is a special and useful shape that looks like a smooth wave that gets smaller and smaller as it moves away from the center. It is defined in two similar ways, but both ways describe the same general idea: it is the ratio of a sine wave to its position. This makes the sinc function very important for understanding waves and signals.

The sinc function is special because it helps us take individual points of a wave and use them to rebuild the whole wave again. This idea is used in many technologies, like in computers and radios, where we need to work with signals that are sampled at regular intervals.

The function was first described in its modern form by Lord Rayleigh when he was studying spherical waves. Because of its unique properties, the sinc function is also known as the cardinal sine function and plays a key role in signal processing and Fourier analysis.

Definitions

The sinc function has two forms: normalized and unnormalized.

In mathematics, the unnormalized sinc function is defined for values other than zero by sinc ⁡ ( x ) = sin ⁡ x / x.

In digital signal processing and information theory, the normalized sinc function is commonly defined for values other than zero by sinc ⁡ ( x ) = sin ⁡ ( π x ) / π x.

For both forms, the value at zero is defined to be 1. The normalization makes the definite integral of the function over all real numbers equal to 1, while the unnormalized sinc function’s integral equals π. The zeros of the normalized sinc function occur at the nonzero integer values of x.

sampling function • digital signal processing • information theory • squeeze theorem • normalization • definite integral • π

Etymology

The sinc function has also been called the cardinal sine or sine cardinal function. The name "sinc" is a short form of its Latin name, sinus cardinalis. This name was first used by Philip M. Woodward and I.L Davies in a 1952 article about information theory and inverse probability in telecommunication. It appeared again in Woodward’s 1953 book Probability and Information Theory, with Applications to Radar.

Properties

The zeros of the sinc function happen at certain points. For the unnormalized sinc, these points are at non-zero whole number multiples of π. For the normalized sinc, the zeros are at non-zero whole numbers.

The sinc function has special points called local extrema. These happen where the sinc function meets the cosine function. This is tied to how the sinc function changes direction.

The normalized sinc function can be written in a special way using an infinite product. It also connects to important math ideas like the gamma function.

The sinc function is very useful in making sense of signals and data. It helps in a process called interpolation, where we fill in missing pieces between known points.

Other facts about the sinc function include its role in solving certain math problems and its connection to other special functions.

Relationship to the Dirac delta distribution

The normalized sinc function can act like a special mathematical tool called a nascent delta function. This means that as a certain value gets very small, the sinc function behaves in a way that matches the properties of the Dirac delta function.

When we use the sinc function in a specific way and let a certain value approach zero, the result relates to the value of another function at zero. This shows how the sinc function can help us understand the Dirac delta function, which is important in many areas of mathematics and science.

Summation

All sums in this section refer to the unnormalized sinc function.

When you add up the values of the sinc function for whole numbers starting from 1 and going forever, the total equals (π – 1/2). This means:

sinc(1) + sinc(2) + sinc(3) + sinc(4) + ... = π – 1/2

If you square each value before adding them up, the total is also π – 1/2:

sinc²(1) + sinc²(2) + sinc²(3) + sinc²(4) + ... = π – 1/2

When the signs of these values alternate (positive, negative, positive, and so on), the total becomes 1/2:

sinc(1) – sinc(2) + sinc(3) – sinc(4) + ... = 1/2

The same alternating pattern works for the squared and cubed values, and both also equal 1/2:

sinc²(1) – sinc²(2) + sinc²(3) – sinc²(4) + ... = 1/2
sinc³(1) – sinc³(2) + sinc³(3) – sinc³(4) + ... = 1/2

Series expansion

The Taylor series helps us understand the sinc function better. For the unnormalized sinc function, it can be written as a special kind of pattern that adds and subtracts smaller parts. This pattern works for any value of x.

The same idea can be used for the normalized sinc function, just with a few extra parts added in.

Higher dimensions

The sinc function can be expanded to work in more than one dimension. When used in two dimensions for a square grid, it is simply the product of two one-dimensional sinc functions. This creates a special pattern in what is called frequency space.

For other types of grids, like a hexagonal pattern, the sinc function is more complex and cannot be created by simply multiplying one-dimensional functions. These special sinc functions can be built using the geometry of shapes called Brillouin zones, which relate to how the grid is organized. This helps create useful tools like the Lanczos window for different multidimensional patterns.

Sinhc

Some writers talk about a related idea called the hyperbolic sine cardinal function. It is defined in a special way: when the value x is not zero, it equals the hyperbolic sine of x divided by x. But when x is exactly zero, the value is simply 1.